What is the derivative of ln((5x^2+x+6)/(x^3)) ?

The derivative of ln((5x^2+x+6)/(x^3)) is (-5x^2-2x-18)/(x(5x^2+x+6))

What is the first derivative of ln((5x^2+x+6)/(x^3)) ?

The first derivative of ln((5x^2+x+6)/(x^3)) is (-5x^2-2x-18)/(x(5x^2+x+6))

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derivative of

ln (\frac{5 x ^{2} + x + 6}{x ^{3}})

\frac{- 5 x ^{2} - 2 x - 18}{x ( 5 x ^{2} + x + 6 )}

Solution steps

\frac{d}{d x} (ln (\frac{5 x ^{2} + x + 6}{x ^{3}}))

Apply the chain rule:

\frac{1}{\frac{5 x ^{2} + x + 6}{x ^{3}}} \frac{d}{d x} (\frac{5 x ^{2} + x + 6}{x ^{3}})

= \frac{1}{\frac{5 x ^{2} + x + 6}{x ^{3}}} \frac{d}{d x} (\frac{5 x ^{2} + x + 6}{x ^{3}})

\frac{d}{d x} (\frac{5 x ^{2} + x + 6}{x ^{3}}) = \frac{- 5 x ^{2} - 2 x - 18}{x ^{4}}

= \frac{1}{\frac{5 x ^{2} + x + 6}{x ^{3}}} \cdot \frac{- 5 x ^{2} - 2 x - 18}{x ^{4}}

Simplify

\frac{1}{\frac{5 x ^{2} + x + 6}{x ^{3}}} \cdot \frac{- 5 x ^{2} - 2 x - 18}{x ^{4}} : \frac{- 5 x ^{2} - 2 x - 18}{x ( 5 x ^{2} + x + 6 )}

= \frac{- 5 x ^{2} - 2 x - 18}{x ( 5 x ^{2} + x + 6 )}

\int 8 x - cos (x) d x

\frac{d}{d x} (2 cos (x))

\frac{d}{d x} ((sin (x^{2}) + x^{2}) (cos (x) - 3 x))

\frac{d y}{d x} = \frac{y ^{2}}{x}

What is the derivative of ln((5x^2+x+6)/(x^3)) ?
The derivative of ln((5x^2+x+6)/(x^3)) is (-5x^2-2x-18)/(x(5x^2+x+6))
What is the first derivative of ln((5x^2+x+6)/(x^3)) ?
The first derivative of ln((5x^2+x+6)/(x^3)) is (-5x^2-2x-18)/(x(5x^2+x+6))